We readily obtain a zig-zag as in (REF ) by applying the functor \(|{-}|\)  to the zig-zag (REF ) and then postcomposing by the evaluation transformation \(e \colon |{-}| \circ \operatorname{Sing}\overset{\sim }{\longrightarrow }1_{\Delta {T}\mathrm {op}}\) .
However, there is an alternative way of obtaining a zig-zag as in (REF ) directly and explicitly, which we think is worth showing:
let \(Q^p\)  be Dugger's cofibrant replacement functor for \({H}_\infty ^p\)  [1]}.
Explicitly, it sends a simplicial presheaf \(F\)  to the two-sided bar construction
\(Q^p F = B^{{H}_\infty }(F,{{C}\mathrm {art}},{{Y}})\,,\) 
